Lattice Models of Polymers: Entanglement Complexity and Confined Geometries

统计力学的晶格模型已被证明是研究聚合物在溶液中的相变行为和平衡性质的有力工具。在这样的模型中，聚合物被认为是由称为单体的重复单元组成的任何长链分子，聚合物链由网格或晶格上的行走表示，因此链中的每个单体与其相邻的单体相距一步。由此产生的晶格聚合物具有灵活性（它可以呈现各种形状），并且很容易结合两个单体不能在同一位置的要求（排除体积效应）。虽然这是一个高度简化的模型，但它对于预测和理解聚合物是大的柔性分子这一事实所导致的现象是有用的。同时，由于大量具有挑战性的开放数学问题，这些模型具有内在的数学兴趣，其中许多问题来自聚合物物理学，最近来自分子生物学。例如，酶作用于DNA以去除缠结，以便正常的细胞过程继续进行。为了研究这一点，可以用一个晶格聚合物链两端连接成一个环的模型来表示DNA分子的大尺度结构，并且我们已经建立了酶对DNA作用的模型。同时，也有兴趣了解聚合物在孔隙或衣壳中的缠结复杂性（如病毒衣壳中的DNA），并对确定缠结在聚合物结晶和聚合物熔体中的作用感兴趣。例如，关于聚合物在衣壳中的包装方式与其缠结复杂性之间的关系，以及对于聚合物熔体，关于测量缠结程度的最佳方法，都存在一些悬而未决的问题。我们还开发了用于研究这些问题的点阵模型。我计划利用组合分析和计算机模拟继续研究聚合物的晶格模型，以便更好地理解酶对DNA的作用和受限聚合物的纠缠复杂性。这些结果将引起分子生物学家和聚合物化学家的兴趣。就酶对DNA的作用而言，更好地了解它们的作用有可能通过改善癌症治疗来影响加拿大人。更一般地说，这些结果将增加我们对聚合物晶格模型和聚合物体系相变的整体理解。

The lattice models of Statistical Mechanics have proved to be powerful tools for studying phase-change behaviour and equilibrium properties for polymers in solution. For such models, a polymer is considered to be any long chain molecule made of repeated units called monomers and a polymer chain is represented by a walk on a grid or lattice so that each monomer in the chain is one step apart from its neighbours. The resulting lattice polymer has flexibility (it can take on a variety of shapes) and the requirement that two monomers cannot be in the same location (the excluded volume effect) is easily incorporated. Although a highly simplified model, it is useful for predicting and understanding  phenomena that result from the fact that polymers are large flexible molecules. At the same time, these models are of intrinsic mathematical interest due to the wealth of challenging open mathematical questions, many of which have arisen from polymer physics and more recently from molecular biology. For example, enzymes act on DNA to remove entanglements in order for normal cellular processes to proceed. To study this, a model in which two ends of a lattice polymer chain are joined into a ring can be used to represent the large-scale structure of the DNA molecule, and models of enzyme action on DNA have been developed by us. At the same time, there is interest in understanding the entanglement complexity of polymers confined in pores or capsids (such as DNA in a viral capsid) and also interest in determining the role of entanglements in polymer crystallization and polymer melts. For example, there are open questions about the relationship between how a polymer is packed in a capsid and its entanglement complexity, and for polymer melts, about the best approach for measuring the extent of entanglement. Lattice models for investigating these questions have also been developed by us. I plan to continue my research on lattice models of polymers using combinatorial analysis and computer simulations in order to better understand enzyme action on DNA and the entanglement complexity of confined polymers. The results will be of interest to molecular biologists and polymer chemists. In the case of enzyme action on DNA, an improved understanding of their action has the potential to impact Canadians by leading to improved cancer treatments. More generally, the results will add to our overall understanding of lattice models of polymers and phase transitions in polymer systems.